The area (in sq. units) of the region described by `{(x ,""y)"":""y^2lt=2x` and `ygeq4x-1}` is : (1) `7/(32)` (2) `5/(64)` (3) `(15)/(64)` (4) `9/(32)`

Options

A. `(7)/(32)`
B. `(5)/(64)`
C. `(15)/(64)`
D. `(9)/(32)`

Answer

D

Explanation

`y^(2)le2x` represents the interior of parabola `y^(2)=2x` (towards focus) and `yge4x-1` represents the half region of plane to the left of the line `y=4x-1`.

`overset(1)underset(-1//2)int((y+1)/(4)-(y^(2))/(2))dy=(1)/(4){(y^(2))/(2)+y}_(-1//2)^(1)-(1)/(6){y^(3)}_(-1//2)^(1)`
`=(1)/(4){((1)/(2)+1)-((1)/(8)-(1)/(2))}-(1)/(6){1+(1)/(8)}`
`=(1)/(4){(3)/(2)+(3)/(8)}-(1)/(6){(9)/(8)}=(15)/(32)-(6)/(32)=(9)/(32)`